Dr J Frost jfrosttiffinkingstonschuk wwwdrfrostmathscom Last modified 4 th May 2016 Objectives Appreciate the purpose of algebraic variables and simplify algebraic expressions ID: 683706
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Slide1
Year 7 Algebraic Expressions
Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com
Last modified:
4
th May 2016
Objectives:
Appreciate the purpose of algebraic variables and simplify algebraic expressions.
Substitute into algebraic expressions.
Form algebraic expressions from worded information.Slide2
INTRO
:: What is algebra?
Source: Google
Algebra concerns representing missing information.
Put simply, we use letters, known as
variables
, to
(usually)
represent numbers.
Usually the value of variables are not initially known, but we hope to combine available information to find their value.
Examples:
might represent someone’s age this year.
might represent an unknown angle.
Slide3
INTRO
:: Examples
of algebraic expressions
Suppose
represented your current age.
What would these expressions represent?
Your age in 4 years time.
Twice your age.
A third of your age.
A few variable naming conventions:
We tend to use a single lower-case letter, either using the English alphabet (a to z) or using the
greek
alphabet (
)
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INTRO
:: Two
stages of algebraic problems
[JMC 2008 Q18] Granny swears that she is getting younger. She has calculated that she is four times as old as I am now, but remember that 5 years ago she was five times as old as I was at that time. What is the sum of our ages now?
Worded problem
Stage 1
: Represent problem algebraically
Let
be my age and
be Granny’s age.
Stage 2
: ‘Solve’ equation(s) to find value of variables.
These next few lessons we’ll be looking at Stage 1.
Stage 2, ‘solving’, we’ll do later this year.
Being able to do these two stages for difficult problems is a vital skill for Maths Challenges/Olympiads.
We won’t solve this now, but how would we approach such a problem?Slide5
Algebraic Simplification – Adding/Subtracting
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How does this ‘simplify’? Why conceptually does it work?
If you had “4 lots of
” and added “3 lots of
”, we’d clearly have “7 lots of
”, i.e.
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More Examples:
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We ‘collected’ the
terms together and the
terms together. We say we ‘collected
like
terms’. Let’s do an activity based on ‘like’ terms.
Bro Note: An algebraic
is written using two back-to-back c’s. Do NOT write as a
symbol.
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x
2
2x2
-3x2
3x3
-
x
3
y
2
5xy
24x
2y
2x2y-1
+
4
x
9
x
5xy
ACTIVITY
::
Collecting Like Terms
Instructions: In pairs, discuss which terms you think might be ‘like’ terms, i.e. they could be combined together into one when adding/subtracting.
Therefore, terms are ‘like terms’ if:
The involve the same variables and powers.
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Quickfire
Examples
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A common Schoolboy
Error
TM
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You might be tempted to simplify to
.
But we saw earlier with BIDMAS that addition and subtraction have the same precedence.
, so we have
lots of
.
Some find it helpful to underline each term (with the + or – symbol on the front) when collecting like terms.
Slide9
ACTIVITY
:: Addition Pyramids
You should have printed the following pyramids. Each block is the sum of the two below it, e.g. as per on the right.
Can you fill in the missing blocks?
2
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1
2
3
4Slide10
Multiplying
In algebra,
we don’t like the
symbol
; instead we put things
next to each other
to indicate they are multiplied.
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Test Your Understanding (so far)
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a
b
c
d
e
Simplify the following.Slide12
Division
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Fractions are ultimately just divisions. How did we simplify this fraction?
Can we apply the same principle to algebraic division?
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Test Your Understanding
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Exercise 1
1
Simplify the following, or write ‘already simplified
’.
a
b
c
d
e
f
g
h
i
j
2
a
c
e
g
i
j
k
l
b
d
f
h
3
a
c
e
g
i
j
b
d
f
h
k
4
a
c
b
d
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Substitution
What is the value of
when
? (Click answer)
12?
36?
We saw by BIDMAS earlier that because indices come first, the
is first squared, THEN multiplied by 3.
means
not
.
If you think of
as “3 lots of
” you’re less likely to make an error.
Slide16
Substitution
If
and
, what is the value of:
Bro Tips:
Start by working out each of the terms first (mentally if you can) leaving the +/- symbols between as they are.
Don’t try to do all at once.
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Terms
?
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Another Example
If
and
, what is the value of:
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Test Your Understanding
If
and
, what is the value of:
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Formulae
A formula (plural: formulae) is a rule to generate one value of interest from others.
For example, the following formula allows you to find the temperature in Fahrenheit given the temperature in Celsius:
The variable of interest goes on the LHS of the equals.
What is
when:
?
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?Slide20
Exercise 2
If
and
what is the value of:
If
, what is:
The smell intensity
given the distance
in metres from the source is given by the formula:
What is the smell intensity when the distance is
metres?
If
, what is:
The profit
of
PippinCo
can be calculated using the number of sold items
and the number of hours open
and the
Calculate the profit if 1420 items are sold and the shop was open 8 hours that day.
If
, what is:
1
a
b
c
d
2
a
b
c
d
e
3
4
a
b
c
d
e
5
6
a
b
c
d
e
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Forming Expressions
[JMC 2008 Q18] Granny swears that she is getting younger. She has calculated that she is four times as old as I am now, but remember that 5 years ago she was five times as old as I was at that time. What is the sum of our ages now?
Worded problem
Stage 1
: Represent problem algebraically
Let
be my age and
be Granny’s age.
Stage 2
: ‘Solve’ equation(s) to find value of variables.
Remember this problem? We’ll be looking how we can turn worded information into algebraic expressions.Slide22
Forming Expressions
Suppose
represents your age. How would you represent:
Your age in 5 years time?
Twice what your age was 5 years ago?
5 years younger than twice your age?
Half what your age was 3 years ago?
Anyone called Bob is four times you age.
Anyone called Charles is two years younger than you.
What is (in terms of
):
The age of one person called Bob:
The total age of a Bob and a Charles:
The total age of you, a Bob and two Charles:
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Later this year you’ll properly learn how to ‘expand brackets’.
?Slide23
A Harder One
“The sum of 5 consecutive whole numbers is 285. What is the smallest of these numbers?”
Supposed we used one variable
. What unknown thing could it represent?
Let
be the smallest number.
Then the five numbers would be:
Then the sum of these numbers would be:
Let
be the middle number.
Then the five numbers would be:
Then the sum of these numbers would be:
Option 1
Option 2
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Why might Option 2 might make the later ‘solving’ stage easier?Slide24
Check Your Understanding
A cat costs £
and a dog £2 less.
What is the cost (in £) of:
4 cats?
3 dogs?
There is a queue of
people. If there are
people in front of me, how many people are behind me?
The average mark of people in a class was 60.
By introducing a suitable variable for the number of people in the class, what would be the total mark of everyone in the class?
Let
be the number of people. Total mark
If a new person joins the class, and gets a mark of 80, what is the total mark now?
If this person’s mark made the average mark rises to 62, give another expression for the total mark of all the people.
A 3 x 3 grid contains nine numbers,
one
in each cell. Each number is doubled to obtain the number on its immediate right and trebled to obtain the number immediately below it.
Use suitable expressions to represent the nine numbers. What expression gives the sum of your numbers?
N
If the sum of the numbers is 13, what is the middle number?
A
B
C
D
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Exercise 3
A number is represented by
. How would we represent:
2 more than the number.
5 times the number.
3 less than
than
twice the number.
Twice as much as 3 less than the number.
A quarter of the number.
or
The cost of a badger is
pence. A racoon is 5 pence more expensive than a badger and a beaver three times as expensive as a badger.
What is the cost of a racoon?
What is the cost of a beaver?
What is the total cost of a racoon and 8 beavers?
You have 7 consecutive numbers, with the smallest number
What are the 7 numbers in terms of
?
Hence what is the sum of all the numbers?
After tennis training, Andy collects twice as many balls as Roger and five more than Maria. If Roger collected
balls, in terms of
, how many balls did:
Andy collect?
Maria collect?
The total number of balls collected?
I think of a number, multiply it by 5, subtract 2, subtract the original number, and then halve it. If the starting number was
, give an expression for the final answer, as simply as possibly.
In
a list of seven consecutive numbers a quarter of the smallest number is five less than a third of the largest number.
If
is the smallest number, find expressions for:
“a quarter of the smallest number”
“five less than a third of the largest number”
1
2
3
4
5
6
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Questions on provided sheet.Slide26
Exercise 3
Pippa thinks of a number. She adds 1 to it to get a second number. She then adds 2 to the second number to get a third number, adds 3 to the third to get a fourth, and finally adds 4 to the fourth to get a fifth number.
If
is the number she started with, what is the sum of her numbers?
Dilly is 7 years younger than Dally. In 4 years time she will be half Dally’s age.
Let Dilly’s age be
and Dally’s
.
Use the first sentence to write a formula for Dilly’s age in terms of Dally’s:
Use the second sentence to write two different expressions for Dilly’s age in 4 years time, one in terms of
and one in terms of
.
A woman had 9 children at regular intervals of 15 months. The oldest is now six times as old as the youngest.
Let
be the age of the youngest child (in years). What is the age of the oldest child (be careful!)
Hence use the information to form an equation relating the ages of the oldest and youngest child (you need not solve it).
Three brothers and a sister shared a sum of money equally among themselves. If the brothers alone had shared the money, then they would have increased the amount they each received by £20.
Suppose the total amount of money is
.
How much money (in terms of
) do the brothers each get if they share just between them?
What expression would represent “an increase of £20 from the previous lower amount they would have got”
7
8
9
10
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[JMO 2007 B1] Find four integers whose sum is 400 and such that the first integer is equal to twice the second integer, three times the third integer and four times the fourth integer.
The four numbers could be represented as
Their sum is
.
Thus
This makes the four numbers
[
JMC 2014 Q20] Box P has
chocolates and box Q has
chocolates, where
and
are both odd and
. What is the smallest number of chocolates which would have to be moved from P to box Q so that box Q has more chocolates than box P?A
B
C
D
E
Solution:
B
Exercise 3
11
12
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